Radon–Nikodym Derivatives as Likelihood Ratios

Summary

The document explains how a likelihood ratio relates to the Radon–Nikodym derivative of one probability measure with respect to another. When both continuous distributions have densities relative to Lebesgue measure, the derivative is the ratio of their densities. Thus, the familiar likelihood ratio is a special case of the more general measure-theoretic construction.

The same reasoning extends to discrete distributions by using counting measure as the dominating measure, yielding a ratio of probability mass functions. This clarifies that the key distinction is generality: the Radon–Nikodym derivative applies when a suitable dominating measure exists, while density and mass-function ratios are particular representations. The brief answer gives these continuous and discrete cases but does not discuss conditions such as absolute continuity, support mismatches, or applications to trading and statistical estimation.

Key ideas

• A Radon–Nikodym derivative expresses one measure relative to another when the required absolute continuity condition holds.
• For continuous distributions dominated by Lebesgue measure, it equals the ratio of their densities.
• For discrete distributions, counting measure gives the corresponding ratio of probability mass functions.
• A likelihood ratio is a specific case of the more general Radon–Nikodym derivative.

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# How is Radon-Nikodym derivative different from the likelihood ratio?


# How is Radon-Nikodym derivative different from the likelihood ratio?












I see that the Radon-Nikodym derivative is the ratio of probability measures, $dP/dQ$. How is this different, in general, from a likelihood ratio of two continuous distributions? I understand the RN-definition broadly applies for discrete/continuous/mixture densities, but beyond that is there any difference?

## Answer by Taylor (score 1)

https://quant.stackexchange.com/a/53267

If $dx$ is Lebesgue measure, then it dominates both measures because they correspond with continuous random variables, and one of the properties of RN derivatives is $$ \frac{dP}{dQ} = \frac{\frac{dP}{dx}}{\frac{dQ}{dx}}. $$ The numerator is the density of $P$, and the denominator is the density of $Q$. This is the second property on wikipedia.

So yes, the likelihood ratio is just a particular case. If these two measures were for discrete random variables, then you would replace $dx$ with the counting measure, and you would get a ratio of probability mass functions.

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